Unidirectional drying of a suspension of diffusiophoretic colloids under gravity

Recent experiments (K. Inoue and S. Inasawa, RSC Adv., 2020, 10, 15763–15768) and simulations (J.-B. Salmon and F. Doumenc, Phys. Rev. Fluids, 2020, 5, 024201) demonstrated the significant impact of gravity on unidirectional drying of a colloidal suspension. However, under gravity, the role of colloid transport induced by an electrolyte concentration gradient, a mechanism known as diffusiophoresis, is unexplored to date. In this work, we employ direct numerical simulations and develop a macrotransport theory to analyze the advective–diffusive transport of an electrolyte-colloid suspension in a unidirectional drying cell under the influence of gravity and diffusiophoresis. We report three key findings. First, drying a suspension of solute-attracted diffusiophoretic colloids causes the strongest phase separation and generates the thinnest colloidal layer compared to non-diffusiophoretic or solute-repelled colloids. Second, when colloids are strongly solute-repelled, diffusiophoresis prevents the formation of colloid concentration gradient and hence gravity has a negligible effect on colloidal layer formation. Third, our macrotransport theory predicts new scalings for the growth of the colloidal layer. The scalings match with direct numerical simulations and indicate that the colloidal layer produced by solute-repelled diffusiophoretic colloids could be an order of magnitude thicker compared to non-diffusiophoretic or solute-attracted colloids. Our results enable tailoring the separation of colloid-electrolyte suspensions by tuning the interactions between the solvent, electrolyte, and colloids under Earth's or microgravity, which is central to ground-based and in-space applications.


Introduction
Unidirectional drying of a colloidal suspension has been used widely for manufacturing microstructured materials, such as ceramics, electrodes, and photonic crystals. 1-7 A typical experimental setup of unidirectional drying involves depositing a mixture of colloids and a volatile solvent into a microchannel. [8][9][10][11][12][13][14][15][16][17][18][19][20] One end of the channel is connected to a large reservoir which provides a constant supply of the mixture to the channel. Evaporation occurs at the other end of the channel, the drying interface, which opens to the atmosphere. Solvent evaporation induces a ow of the mixture toward the drying interface. The colloids are carried by the solvent and concentrate at the drying interface, forming a colloidal lm. Recent experiments 19 and simulations 21 demonstrated that gravity plays an important role in the phase separation process. Specically, under evaporation of a nonelectrolyte-colloid suspension, colloid concentration increases on approaching the drying interface. Sedimentation of colloids causes a backow of the mixture away from the drying interface, which enables a continuous growth of the colloidal lm.
Here, we hypothesize that phase separating an electrolytecolloid suspension could be drastically different from that of a non-electrolyte-colloid suspension due to a mechanism known as diffusiophoresis. [22][23][24][25][26] Diffusiophoresis refers to the deterministic motion of particles induced by a surrounding concentration gradient of solute. Diffusiophoresis has received much attention in recent years for its ability to manipulate colloid transport in a wide range of applications, including mixing and separation,  enhanced oil recovery, [50][51][52] and drug delivery. 53,54 In our hypothesis, we envision that evaporation will induce an electrolyte concentration gradient, by the same token as that of the colloid, where the electrolyte concentration will increase toward the drying interface. The electrolyte gradient will in turn induce diffusiophoretic motion of colloids, which will drastically alter the colloid transport. The diffusiophoretic velocity of a colloid is given by V = MVlog S, [22][23][24][25][26] where S is the ionic solute concentration and the mobility M encompasses information of the electrolyte and colloid such as the ion valence and colloid surface potential. The mobility can be positive or negative, corresponding to diffusiophoresis driving colloids up (solute-attracted) or down (solute-repelled) the solute gradient, respectively. The diffusiophoretic velocity (∼10 −6 m s −1 ) 25,34 is typically comparable to or orders of magnitude larger than the evaporation-induced uid ow that carries the colloids in a drying cell (∼10 −9 to 10 −6 m s −1 ). 4,7,55 This strengthens our hypothesis that the phase separation of an electrolyte-colloid suspension could be drastically different from that of a non-electrolyte-colloid suspension.
In this work, we utilize direct numerical simulations and develop a macrotransport theory to analyze the advectivediffusive transport of an electrolyte-colloid suspension in a unidirectional drying cell. The electrolyte and colloid motion are inuenced by diffusiophoresis, gravity, and solvent evaporation. We report three key ndings that conrm our hypothesis. First, there is a strong phase separation in drying a suspension of solute-attracted colloids, which generates the thinnest colloidal layer relative to drying a suspension of nondiffusiophoretic or solute-repelled colloids. Second, when colloids are solute-attracted or weakly solute-repelled, gravity could affect the colloid transport and thickness of the colloidal layer substantially. However, when colloids are strongly soluterepelled, diffusiophoresis could nullify the effect of gravity on colloid transport by eliminating the formation of a signicant colloid concentration gradient. Third, our macrotransport theory predicts new early-time and long-time scalings of the growth of the colloidal layer which agree with direct numerical simulations. The colloidal layer generated by solute-repelled colloids could be ten times thicker than that by nondiffusiophoretic colloids.
The rest of this article is outlined as follows. In Section 2, we formulate the problem by presenting the governing equations and boundary conditions for the transport of the solvent, ionic solute, and colloids. Derivations of the macrotransport theory of diffusiophoretic colloid transport under varying strengths of gravity as well as scalings of the growth of the colloidal layer are presented in Appendix A. In Section 3, we present our results and elaborate on the three above-mentioned key ndings. In Section 4, we summarize this study and offer ideas for future work.

Problem formulation
Consider a channel that consists of two parallel plates of length L separated by a distance H (Fig. 1). Initially, the channel is lled uniformly with a dilute suspension of constant density, r i , comprising a volatile solvent of kinematic viscosity n s , a nonvolatile ionic solute of concentration S i , and non-volatile colloids of concentration C i . The le-end of the channel is connected to a large reservoir of the suspension. Evaporation induces a ow of the suspension with a constant velocity at the drying interface, the right-end of the channel. A colloidal layer of thickness D is formed at the drying interface.
When colloids and solute concentrate on approaching the drying interface, the density of the mixture increases. The local density of the mixture, r, is related to the local concentration of the colloid, C, and solute, S, via 21,56 where b c and b s are the solutal expansion coefficient of the colloid and the solute, respectively. A difference in the density induces a gravitational body force. Under the Boussinesq approximation for microscale ows, 21,[56][57][58][59] this gravitational force appears in the Stokes equation that governs the uid motion, along with the continuity equation where U is the solvent ow velocity, P is the pressure deviation from the initial hydrostatic pressure, and g is the gravitational acceleration. The evolution of the solute concentration is governed by the advection-diffusion equation where T is time and D s is the solute diffusivity. A solute concentration gradient is developed over time, which induces a diffusiophoretic velocity of the colloid, V = MVlog S. [22][23][24][25][26] The evolution of the colloid concentration is governed by the advection-diffusion equation which comprises the diffusiophoretic velocity 39-41 where D c is the colloid diffusivity. Following prior work, the colloidal layer thickness, D, and the mean position of the colloid distribution, U, are dened as 21,42 and The initial and boundary conditions that accompany eqn (1)-(4) are as follows. The initial conditions at T = 0 are C = C i and S = S i .
For the boundary conditions, at the le-end of the channel, X = −L, connection to a large reservoir of suspension requires that C = C i and S = S i . At the channel walls, Z = 0 and Z = H, no hydrodynamic slip and no penetration of the solvent require that Diffusioosmosis adjacent to the channel walls is ignored in the present study to highlight the effect of diffusiophoresis. In practice, diffusioosmosis can be mitigated by precoating the channel walls with a mono-molecular layer of non-cross-linked polyacrylamide. 60,61 No penetration of the colloids and solute requires that vC vZ ¼ 0 and vS vZ ¼ 0: At the drying interface, X = 0, it requires that vU Z vX ¼ 0; where eqn (11) represents that the solvent velocity in the Xdirection, U X , equals the evaporation rate E, 21 eqn (12) represents that the drying interface is a free surface, and eqn (13) ensures the non-volatility of the solute and colloids. We introduce the following non-dimensionalization scheme, Upon non-dimensionalization, eqn (1)-(4) become Five dimensionless groups emerge. These include two Rayleigh numbers of the colloids and the solute, Ra c = b c gH 3 C i /(n s D s ) and Ra s = b s gH 3 S i /(n s D s ), which describe the relative strength between gravity and diffusion; a Peclet number, Pe = EH/D s , which describes the relative strength between solvent convection and solute diffusion; the ratio of the colloid to solute diffusivity, D c /D s ; and the ratio of the diffusiophoretic mobility

Results and discussion
In this section, we examine the time evolution of the xcomponent solvent velocity u x , solute concentration s, xcomponent diffusiophoretic velocity v x , colloid concentration c, colloidal layer thickness d, and the mean position of the colloid distribution u. We have conducted all simulations with a channel of length much larger than the channel height, l = 10 3 , so that key ow features developed near the drying interface, e.g., uid backow, are not hindered by the presence of the mixture reservoir. In all simulations, the maximum colloid volume fraction C < 0.05 (maximum c < 500) so that particleparticle interactions are negligible. 34 We start by showing the impact of varying strengths of diffusiophoresis (M/D s ) in Section 3.1. This is followed by showing the impact of varying strengths of gravity (Ra c and Ra s ) in Section 3.2.
3.1 Impact of varying strengths of diffusiophoresis 3.1.1 Velocity and concentration elds. Let us rst examine Fig. 2(a) which is obtained with M/D s = 0 and corresponds to no diffusiophoresis in the system. This recovers the key observations in prior work 21 and validates our simulation framework. We state and explain the observations as follows. At early times, t = 1, evaporation induces a net solvent ow to the right across any cross section of the channel, as prescribed by the boundary condition at the drying interface. Indeed, shown in the second and fourth panel, the solvent ow carries the solute and colloids to the right and they are accumulating near the drying interface. However, the colloid (vc/vx) and solute (vs/vx) concentration gradients are not signicant near the drying interface and hence there is no uid backow. This can be understood by the scaling of the solvent backow velocity, which is obtained from a balance between the viscous and buoyancy terms in eqn (15) and (16). 21,67 That is, u x,back / 0 as vc/vx / 0 and vs/vx / 0. As a result, the parabolic ow prole resembles a pressure-driven ow with the maximum velocity at the centerline of the channel (z = 1/2) and zero velocity at the channel walls (z = 0 and z = 1) due to no hydrodynamic slip. Going from t = 1 to 10 2 in Fig. 2(a), solvent ow continues to carry colloids and solute towards the drying interface. The colloid and solute concentration gradients near the drying interface strengthen. Hence, the parabolic ow prole weakens at t = 10 2 . Instead, colloids undergo sedimentation and induce a backow of the suspension as shown in the rst panel where, in the bottom half of the channel, the suspension ows to the le as indicated by a negative u x . As time goes by at t = 10 4 , the colloid and solute concentration gradients near the drying interface lengthen in the x-direction and continue to strengthen. As a result, the solvent ow prole develops fully, with a ow toward and away from the drying interface in the upper-half and lower-half of the channel, respectively. The backow increases in magnitude according to eqn (20). Note that, during evolution, the colloid concentration prole, c, is increasingly asymmetric about the centerline of the channel, which follows from the asymmetry of the solvent ow.
As an overview of Fig. 2(a), the diffusiophoretic velocity, v x , is zero everywhere at all times, conrming that the solute gradient induces no diffusiophoresis to colloids due to the present case of M/D s = 0 (recall that V = MVlog S). On a different note, distinct from the asymmetric colloid distribution, c, about the channel centerline due to the solvent backow, the solute distribution, s, is symmetric. This can be understood by examining the relation H 2 /D s = Pe(H/E) with Pe = 10 −2 in the present case. Physically, the solute takes a much shorter time to diffuse across the channel height (H 2 /D s ) compared to it being transported by the backow (H/E) for a unit distance in the x-direction (remember that x = X/H). In other words, diffusion has made uniform the solute distribution across the channel height before the distribution gets distorted by the backow in the xdirection. Hence, the solute concentration is uniform in the zdirection at any position x.
Next, let us look at Fig. 2(b) that is obtained with M/D s = 0.5 and corresponds to solute-attracted diffusiophoresis in the system. The presence of solute-attracted diffusiophoresis is conrmed by a positive diffusiophoretic velocity, v x (up the solute gradient to the right), at all times. At an early time, t = 1, comparing Fig. 2(b) and (a), the solvent ow velocity u x in panel (b) and (a) are identical. Physically, this means that diffusiophoresis does not alter the solvent ow so long as solvent backow is absent. Furthermore, the colloid concentration, c, and concentration gradient, vc/vx, in Fig. 2(b) are higher than those in Fig. 2(a). This is because, under solute-attracted diffusiophoresis, there is an additional diffusiophoretic velocity of the colloids whose direction is up the solute gradient to the right, transporting more colloids toward the drying interface compared to no diffusiophoresis.
At long times, t = 10 2 and t = 10 4 , comparing Fig. 2(b) and (a), the magnitude of u x and c in panel (b) are higher than those in panel (a) at the same time t, although the proles of u x , s, and c between panel (b) and (a) at the same t are qualitatively the same. These observations can be understood as follows. As noted above, solute-attracted diffusiophoresis increases c and vc/vx near the drying interface. The solvent (backow) velocity also increases according to eqn (20). On a different note, a colloidal layer of high colloid concentration is formed near the drying interface and is thinner than that in Fig. 2(a) with no diffusiophoresis, meaning that solute-attracted diffusiophoresis causes strong phase separation.
Next, let us examine Fig. 2(c) that is obtained with M/D s = −0.5 and corresponds to weakly solute-repelled diffusiophoresis. The phenomena demonstrated in Fig. 2(c) are the opposite of Fig. 2(b). First, the presence of solute-repelled diffusiophoresis is conrmed by a negative diffusiophoretic velocity, v x (down the solute gradient to the le), at all times. At long times, t = 10 2 and t = 10 4 , comparing Fig. 2(c) and (a), the magnitude of u x and c in panel (c) are lower than those in panel (a) at the same time t, although the proles of u x , s, and c between panel (c) and (a) at the same t are qualitatively the same. These observations can be understood as follows. Soluterepelled diffusiophoresis induces a convective ux of colloids down the solute gradient to the le, which partially cancels the convective ux of colloids up the solute gradient to the right due to the evaporation-induced solvent ow. This leads to an overall weaker transport of colloids toward the drying interface in Fig. 2(c) compared to Fig. 2(a). As a result, the colloid concentration and concentration gradient decrease, and hence the backow velocity decreases according to eqn (20). Also, note that the colloidal layer formed near the drying interface is thicker than that in Fig. 2(a), meaning that solute-repelled diffusiophoresis weakens phase separation.
Next, let us look at Fig. 2(d) that is obtained with M/D s = −1 and corresponds to strongly solute-repelled diffusiophoresis. The presence of strongly solute-repelled diffusiophoresis is conrmed by a more negative diffusiophoretic velocity, v x , relative to Fig. 2(c). Notably, at t $ 10 2 , the solvent ow prole, u x , and colloid distribution, c, are qualitatively different than all previous cases in Fig. 2(a)-(c). Specically, solvent backow no longer exists. This is because the diffusiophoretic ux of colloids to the le is strong enough to counter a signicant portion of that due to solvent convection to the right. As a result, the colloid concentration is nearly uniform, where the maximum and minimum values of c are 2.4 and 1, respectively, even at t = 10 4 . Therefore, the colloid concentration gradient built up is too weak to generate a solvent backow. Phase separation is the weakest in this case, resulting in the thickest colloidal layer. In the next section, we quantify the time evolution of the colloidal layer thickness and mean position of the colloid distribution in the above cases.
3.1.2 Colloidal layer thickness and mean colloid position. Fig. 3(a) shows the time evolution of the colloidal layer thickness, d, for different non-positive M/D s , with D c /D s = 10 −2 , Pe = 10 −2 , Ra c = 1.2, and Ra s = 4 × 10 −2 . In the following, we analyze the results and highlight the scalings of the growth of the colloidal layer thickness. We remark that the scalings obtained from direct numerical simulations in Fig. 3(a) agree with those obtained from a macrotransport theory. Readers are referred to Appendix A for detailed derivations of the macrotransport theory.
Let us start by analyzing the case with no diffusiophoresis (dashed line) in Fig. 3(a). The colloidal layer thickness grows diffusively as ffiffi t p at early times and grows as t 0.4 at long times due to a balance of convection and gravity [eqn (37) and (39)]. This recovers the results of prior work. 21 Next, let us analyze the cases with solute-repelled diffusiophoresis, shown by solid lines in Fig. 3(a). Regardless of the strengths of diffusiophoresis, M/D s , our simulations show a new early-time scaling, where d grows as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi À2M=D s p ffiffi t p . Our macrotransport theory recovers this scaling and identies that this scaling is due to a balance between transient and diffusiophoretic transport of colloids [eqn (33)]. The prefactor of the scaling, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi À2M=D s p , also correctly predicts the trend shown in Fig. 3(a) where, at a xed t, d increases as M/D s becomes more negative.
Diffusiophoresis also alters the long-time scaling of d. In Fig. 3(a), our simulations show that the scaling transition from t 0.4 at M/D s = 0 to a plateau at M/D s = −0.25. Our macrotransport theory recovers this plateau and shows that it is due to a balance between uid convection and diffusiophoresis [eqn (34)]. The plateau persists up to M/D s = −0.75. However, for strong diffusiophoresis where M/D s = −1, d grows continuously and deviates from the plateau. We identify that this deviation is due to a phenomenon that the peak of the colloid distribution is transported away from the drying interface, as shown in Fig. 4(a). This phenomenon is unique for systems with strongly solute-repelled diffusiophoresis, which is in contrast to systems with weakly solute-repelled diffusiophoresis (and soluteattracted diffusiophoresis or no diffusiophoresis) where the peak of the colloid distribution stays at the drying interface at all times, as shown in Fig. 4(b). As an overview of Fig. 3(a), the colloidal layer generated by solute-repelled diffusiophoretic colloids could be an order of magnitude thicker than that by non-diffusiophoretic colloids, highlighting the impact of diffusiophoresis on the production of colloidal lms.  For the cases with solute-attracted diffusiophoresis, d is not applicable to quantify the colloid distribution because the colloid concentration C at some positions are smaller than the initial colloid concentration C i , leading to a negative d which is physically irrelevant [eqn (5)]. To still quantify the colloid distribution, in Fig. 3(b) we show the mean position of the colloid distribution, u, for different M/D s . Note that u is also applicable to quantify cases with solute-repelled diffusiophoresis and we show their data in Fig. 3(b) for comparison. Fig. 3(b) shows that, at a xed t, u approaches x = 0 monotonically as M/D s becomes more positive. Physically, as M/D s increases from negative to positive, the colloid diffusiophoretic velocity v x switches from directed-away to directed-toward the drying interface where the solute accumulates, transporting more colloids toward the drying interface. Thus, the mean position of the colloid distribution shis toward the drying interface at x = 0.
3.2 Impact of varying strengths of gravity 3.2.1 Velocity and concentration elds. Let us rst examine Fig. 5(a) which is obtained with no diffusiophoresis, M/D s = 0. In the absence of gravitational effect, Ra c = Ra s = 0, colloid (vc/vx) and solute (vs/vx) concentration gradients do not cause solvent backow according to eqn (20). Thus, the solvent ow prole, u x , remains parabolic, with the maximum velocity along the channel centerline and zero velocity at the channel walls. As Ra c and Ra s become non-zero, gravity causes sedimentation of colloids and induces a backow of the suspension, indicated by a negative u x . As Ra c and Ra s continue to increase in the third set of density proles in Fig. 5(a), the maximum u x and c, which occur near the drying interface, increases and decreases, respectively. This can be understood as follows. According to eqn (20), the magnitude of the backow u x,back increases as Ra c and Ra s increase. A larger backow carries more colloids to the le and hence decreases the maximum c near the drying interface. Note that, the back-ow also weakens the colloid and solute concentration gradients which in turn has a weakening effect on the solvent backow. However, the weakening of the backow induced by decreasing the colloid concentration gradient is smaller than the strengthening of the backow due to increasing Ra c and Ra s . As a result, from eqn (20), overall the backow velocity increases as Ra c and Ra s increase.
Next, let us look at Fig. 5(b) which is obtained with soluteattracted diffusiophoresis, M/D s = 0.5. In the absence of gravitational effect, Ra c = Ra s = 0, comparing Fig. 5(b) and (a), the solvent velocity u x in panel (b) and (a) are identical whereas the colloid concentration c and concentration gradient vc/vx in (b) are higher than those in (a). In the presence of gravity, Ra c = [0.12, 1.2] and Ra s = [4 × 10 −3 , 4 × 10 −2 ], the magnitude of u x and c in panel (b) are higher than those in panel (a) at the same time t, although the proles of u x , s, and c between panel (b) and (a) at the same t are qualitatively the same. Here, the physical explanations are the same as those in comparing Fig. 2(b) and (a) and we do not repeat them.
Next, let us examine Fig. 5(c) which is obtained with weakly solute-repelled diffusiophoresis, M/D s = −0.5. When Ra c = Ra s = 0, comparing Fig. 5(c) and (a), u x in panel (c) and (a) are identical whereas c and dc/dx in (c) are lower than those in (a). When Ra c = [0.12, 1.2] and Ra s = [4 × 10 −3 , 4 × 10 −2 ], the magnitude of u x and c in panel (c) are lower than those in panel (a) at the same time t, although the proles of u x , s, and c between panel (b) and (a) at the same t are qualitatively the same. Again, the physical explanations here are identical to those in comparing Fig. 2(c) and (a) and we do not repeat them.
Next, let us look at Fig. 5(d) which is obtained with strongly solute-repelled diffusiophoresis, M/D s = −1. Notably, all density proles are almost invariant under different Ra c and Ra s . We understand this by recalling Fig. 2(d) that strongly soluterepelled diffusiophoresis prevents the formation of a strong colloid concentration gradient, vc/vx / 0. Thus, according to eqn (20), solvent backow is absent. It follows that the evaporation-induced parabolic ow prole persists, regardless of the value of Ra c and Ra s . On a different note, comparing to Fig. 5(a)-(c), strongly solute-repelled diffusiophoresis in Fig. 5(d) leads to the weakest phase separation and develops the thickest colloidal layer. The colloidal layer formed in Fig. 5(d) is also the most uniform across the z-direction among all cases. In the next section, we quantify the time evolution of the colloidal layer thickness and mean position of the colloid distribution in the above cases.
3.2.2 Colloidal layer thickness and mean colloid position. Let us rst examine the case with no diffusiophoresis. Fig. 6(a) shows the time evolution of the colloidal layer thickness, d, for different Ra c and Ra s , with M/D s = 0, D c /D s = 10 −2 , and Pe = 10 −2 . Here, the green line with strong gravitational effects, Ra c = 1.2 and Ra s = 4 × 10 −2 , is identical to the dashed line in Fig. 3(a), where d grows as ffiffi t p at early times and t 0.4 at long times. As gravitational effects weaken, represented by a decrease in Ra c and Ra s , the early-time diffusive scaling persists. However, the long-time scaling weakens and eventually reaches a plateau in the limit of Ra c = Ra s = 0 [eqn (38)]. In sum, in the absence of diffusiophoresis, while gravity has negligible effects on electrolyte-colloid phase separation and thus the colloidal layer thickness at early times, an increasing gravitational effect weakens phase separation and develops a thicker colloidal layer at long times. These results recover the key ndings in prior work. 21 Next, let us analyze the case with solute-attracted diffusiophoresis in Fig. 6(b). As noted in Section 3.1.2, the traditional denition of d is not applicable to quantify the transport of solute-attracted diffusiophoretic colloids but the mean position of the colloid distribution, u, could be measured instead. Fig. 6(b) shows the time evolution of u for different Ra c and Ra s , with M/D s = 0.5, D c /D s = 10 −2 , and Pe = 10 −2 . Fig. 6(b) shows that u becomes more negative as Ra c and Ra s increase.
Physically, larger Ra c and Ra s imply a larger backow [eqn (20)], which transports more colloids away from the drying interface. This shis the mean position of the colloid concentration distribution to the le and therefore u becomes more negative. In other words, similar to the case with no diffusiophoresis, in the presence of solute-attracted diffusiophoresis, gravity has negligible effects on electrolyte-colloid phase separation and thus the colloidal layer thickness at early times. However, an increasing gravitational effect leads to a stronger solvent back-ow, which weakens phase separation and develops a thicker colloidal layer at long times.
Next, let us analyze the case with weakly solute-repelled diffusiophoresis, M/D s = −0.5, in Fig. 6(c). Here, the green line with strong gravitational effects, Ra c = 1.2 and Ra s = 4 × 10 −2 , is identical to the red line in Fig. 3(a), where d grows as ffiffi t p at early times and plateaus at long times. As gravitational effects weaken, the early-time diffusive scaling persists. However, the long-time scaling grows and eventually becomes t 0.37 in the limit of Ra c = Ra s = 0. We note that the origin of the t 0.37 scaling is different from the similar t 0.4 scaling in Fig. 6(a). Specically, the t 0.37 scaling of the dashed line in Fig. 6(c) is associated with a system with diffusiophoresis but no gravitational effects, whereas the t 0.4 scaling of the dashed line in Fig. 6(c) is associated with a system under gravity but without diffusiophoresis. Our simulations show that the t 0.37 scaling is still evolving at t = 10 6 but the t 0.4 scaling is reached and invariant at t $ 10 4 . On a different note, due to the competition between colloid transport induced by gravity and diffusiophoresis, increasing gravity strengthens phase separation and decreases d in the presence of diffusiophoresis [ Fig. 6(c)] whereas increasing gravity weakens phase separation and increases d in the absence of diffusiophoresis [ Fig. 6(a)]. This demonstrates another qualitative impact of diffusiophoresis on unidirectional drying, in addition to the order-of-magnitude enhancement in d exhibited in Fig. 3(a). Lastly, we show the time evolution of d with strongly soluterepelled diffusiophoresis, M/D s = −1, in Fig. 6(d). The green line with strong gravitational effects, Ra c = 1.2 and Ra s = 4 × 10 −2 , is identical to the green line in Fig. 3(a). Here, the data for different Ra c and Ra s overlaps onto the same line, meaning that under strongly solute-repelled diffusiophoresis gravity has no effect on phase separation and therefore d. This is due to the absence of solvent backow as explained in Fig. 5(d). As an overview of Fig. 6(a)-(d), the effect of gravity on d is the most prominent when diffusiophoresis is absent [ Fig. 6(a)].

Conclusions
In this work, we have utilized direct numerical simulations and developed a macrotransport theory to quantify the advectivediffusive transport of diffusiophoretic colloids in a unidirectional drying cell. We focus on analyzing the time evolution of the solvent velocity u x , solute concentration eld s, colloid diffusiophoretic velocity v x , colloid concentration eld c, colloidal layer thickness d, and mean position of the colloid distribution u.
The rst part of our analyses focuses on the impact of varying diffusiophoresis under constant, non-zero gravity. At long times, as the colloids switch from solute-attracted to soluterepelled (M/D s becomes more negative), the magnitude of u x and c near the drying interface decreases. Solvent backow is absent in a suspension of strongly solute-repelled colloids, since diffusiophoresis prevents the formation of a strong colloid concentration gradient. We further quantify d. The scalings of d obtained from our macrotransport theory agree with simulations. For weakly solute-repelled colloids, d grows diffusively at early times and plateaus at long times. For strongly solute-repelled colloids, d also grows diffusively initially but continues to grow at long times. The colloidal layer thickness of solute-attracted colloids cannot be quantied by the traditional formula for non-diffusiophoretic colloids. Thus, we compute the mean position of the colloid distribution and show that diffusiophoresis concentrates solute-attracted colloids near the drying interface where the solute accumulates. Overall, phase separation is the strongest and weakest with solute-attracted and solute-repelled colloids, respectively. The colloidal layer formed by solute-repelled colloids could be ten times thicker than that by non-diffusiophoretic colloids.
The second part of our analyses focuses on the impact of varying gravity at long times. In the absence of gravity, u x is independent of the strength of diffusiophoresis and the colloid concentration near the drying interface decreases as the colloids switch from solute-attracted to solute-repelled. In the presence of constant non-zero gravity, as M/D s becomes more negative, the magnitude of u x and c near the drying interface decreases. A suspension of strongly solute-repelled colloids is a special case. Strongly solute-repelled diffusiophoresis prevents the formation of a signicant colloid concentration gradient and the subsequent solvent backow. As a result, all four quantities u x , s, v x , and c are invariant regardless of the strength of gravity. We further quantify d and u. Increasing gravity is shown to weaken phase separation and increase d in the absence of diffusiophoresis whereas gravity has the opposite effect on phase separation and d in the presence of weakly solute-repelled diffusiophoresis. For strongly solute-repelled colloids, gravity has no effect on phase separation and d.
The present work considers dilute colloidal suspensions and highlights the important role of diffusiophoresis in colloidal lm formation. For future work, the present model could be extended to consider channel walls with non-uniform electrokinetic properties 70-73 as well as converging or diverging channels. 44 We expect that these factors will have qualitative impacts on the thickness and uniformity of the colloidal layer. To quantify the formation of a dense colloidal lm, one could turn to particle dynamics simulations that account for the nite size of particles. Some recent work has been done in this direction [74][75][76][77][78][79] but, to the authors' knowledge, the electrophoretic component of diffusiophoresis has been ignored. It will be of interest to conduct particle dynamics simulations that include complete diffusiophoresis and compare with present results for the development of a reduced-order model.
Appendix A: a macrotransport theory for diffusiophoretic colloids under gravity In this section, we derive a macrotransport theory that predicts the growth of the colloidal layer thickness, d, presented in Sections 3.1.2 and 3.2.2. In the following, we start with deriving the theory for an electrolyte-colloid suspension under gravity with diffusiophoresis, which is a novel result of this work. Then, we will show that our theory could reduce to that for a nonelectrolyte-colloid suspension under gravity with no diffusiophoresis developed in prior work. 21 To derive the macrotransport theory for an electrolyte-colloid suspension under gravity with diffusiophoresis, rst we recall the two-dimensional colloid transport eqn (19) vc vt þ Following prior work in macrotransport theory, 80,81 the colloid concentration eld, c, is written as in terms of its cross-sectional average, c 0 = hci, and variation from the average, Pec 1 , c(x,z,t) = c 0 (x,t) + Pec 1 (x,z,t), where Pec 1 ( c 0 and the cross-sectional average is hð$Þi ¼ Ð 1 0 ð$Þ dz. Substituting eqn (22) into (21) and performing cross-sectional averaging gives The objective now is to obtain u x and c 1 and then substitute them into eqn (23).
To obtain u x , we invoke the continuity equation that gives u z ∼ u x /d and assume d [ 1 so that u x [ u z z 0 and vu x /vx z 0. Using these conditions in eqn (15) and (16) The ow eld u x comprises a pressure-driven ow induced by solvent evaporation, u p x , and a ow induced by gravity, u g x , 82 i.e., u x = u p x + u g x . By linearity of the equations, u p x and u g x can be obtained as